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The math of reshuffling – Gloomhaven decks

As promised in my last post, I want to talk briefly about how reshuffling of decks impacts the average damage in Gloomhaven.

The intuitive (but wrong) answers

If you ignore reshuffling, then for each card in a 20-card deck (like Gloomhaven’s) should have a simple 1/20 = 5% chance of being drawn. So, there should be a 5% chance of getting a 2x card or a Miss card in Gloomhaven, right? This just make sense, and also resonates with our fond feelings for d20 rolls in tabletop games! Sadly, it’s wrong. Reshuffling those 2x and Miss cards changes things!

The right answers

So, how does reshuffling a card impact the odds of drawing it? Obviously it should increase the odds, if anything.

Think about it this way: If you reshuffle a deck each time you encounter a card, how far through the deck (on average) will you progress before you encounter that card?

If you have just one card that makes you reshuffle, it will be equally likely to be at the top, middle, end, or any other place in the deck. So, on average, you will encounter it halfway through the deck.

Likewise, if you have 2 cards that force you to reshuffle, then on average you will see one of these cards when you get 1/3 of the way through the deck.

If this is hard to visualize, I sometimes think of it as what would happen if I set up a gazillion decks and went through them all. Sometimes the first card would be a reshuffle, sometimes I’d almost finish the deck before reshuffling. But, on average I’d get 1/3 of the way through. It makes sense if you consider that the average way to distribute these 2 reshuffle cards in your deck is to put one 2/3 of the way down, and the other 1/3 of the way down in your deck.

Phew!

Why does this matter?

Take the example of only having one reshuffle card in your 20-card deck. If you see this card on average about halfway through, then you see it twice as often since you reshuffle when you see it. Think of just going through that “average” deck. Every 9-10 cards you see this one particular card and reshuffle again. That matters!

So you see the card twice as often! The odds of drawing it are 2/20 = 10%, right?

Blasphemy.

That would mean the odds of one card are 2/20, and the odd of other cards add to 19/20. You can tell this doesn’t add up. Literally.

Having one reshuffle card really makes the deck equivalent to having 21 cards, which … sounds weird, I know. The odds of every normal card works out to 1/21 and the odds of the reshuffle card are 2/21. Strange, yes – but when you consider cycling through an “averagely distributed” deck as discussed earlier, then a cycle would consist of ~9.5 normal card, reshuffle card, 9.5 normal, reshuffle. That’s 21 cards, and 2 of them were reshuffle!

So in the long term, the odds of drawing that reshuffle card are 2/21 = ~9.524%, and the odds of drawing each other individual card are 1/21 = ~4.762%.

Uh, remember that Gloomhaven game?

Right! So the standard deck in Gloomhaven has not one, but two reshuffle cards. Using the same arguments as above, I claim that this makes the odds of each reshuffle card 1.5/21 = ~7.143%.

I’ll explain. With two reshuffle cards, you see one every 1/3 of the way through the deck. This is equivalent in the long term to having 21 cards, where 3 of them are reshuffle cards. In essence, your cycle through an average deck is: pull 6.5 normal cards, pull a reshuffle card, pull 6.5 normal, then a reshuffle, 6.5 more normal cards, and a reshuffle. That’s 21 cards with 3 of the cards you encounter being reshuffle cards. So, your odds of a reshuffle card are 3/21, and you are equally likely to pull the 2x or the Miss card, so each will split the probability evenly – 1.5/21 odds for each!

5 thoughts on “The math of reshuffling – Gloomhaven decks”

Interesting perspective – it never occured to me that it works this way.
But I would argue that for a player it is more important whats the
probability distribution on his next attack than a mean frequency of drawing ‘Miss’.
So yeah you will draw misses 7% of your attacks (ignoring multi targeting). But still you have only 1/20 on your first draw, 1/19 on second and so on. That is until you reshuffle and reset your deck (1/20 again).

Yes, I agree. I should have stressed more that I was considering “long-term” average, in a sense. Not sure how to phrase it. Your comment makes me realize that I should write about card-counting in Gloomhaven, since I think that would be neat!